NAG CL Interface
f07mnc (zhesv)

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1 Purpose

f07mnc computes the solution to a complex system of linear equations
AX=B ,  
where A is an n×n Hermitian matrix and X and B are n×r matrices.

2 Specification

#include <nag.h>
void  f07mnc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, Complex a[], Integer pda, Integer ipiv[], Complex b[], Integer pdb, NagError *fail)
The function may be called by the names: f07mnc, nag_lapacklin_zhesv or nag_zhesv.

3 Description

f07mnc uses the diagonal pivoting method to factor A as
order uplo A
Nag_ColMajor Nag_Upper U D UH
Nag_ColMajor Nag_Lower L D LH
Nag_RowMajor Nag_Upper UH D U
Nag_RowMajor Nag_Lower LH D L
where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1×1 and 2×2 diagonal blocks. The factored form of A is then used to solve the system of equations AX=B.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: uplo Nag_UploType Input
On entry: if uplo=Nag_Upper, the upper triangle of A is stored.
If uplo=Nag_Lower, the lower triangle of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: n Integer Input
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
4: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
5: a[dim] Complex Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
On entry: the n×n Hermitian matrix A.
If order=Nag_ColMajor, Aij is stored in a[(j-1)×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[(i-1)×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: if fail.code= NE_NOERROR, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A=UDUH, A=LDLH, A=UHDU or A=LHDL as computed by f07mrc.
6: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraint: pdamax(1,n).
7: ipiv[dim] Integer Output
Note: the dimension, dim, of the array ipiv must be at least max(1,n).
On exit: details of the interchanges and the block structure of D. More precisely,
  • if ipiv[i-1]=k>0, dii is a 1×1 pivot block and the ith row and column of A were interchanged with the kth row and column;
  • if uplo=Nag_Upper and ipiv[i-2]=ipiv[i-1]=-l<0, (di-1,i-1d¯i,i-1 d¯i,i-1dii ) is a 2×2 pivot block and the (i-1)th row and column of A were interchanged with the lth row and column;
  • if uplo=Nag_Lower and ipiv[i-1]=ipiv[i]=-m<0, (diidi+1,idi+1,idi+1,i+1) is a 2×2 pivot block and the (i+1)th row and column of A were interchanged with the mth row and column.
8: b[dim] Complex Input/Output
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×nrhs) when order=Nag_ColMajor;
  • max(1,n×pdb) when order=Nag_RowMajor.
The (i,j)th element of the matrix B is stored in
  • b[(j-1)×pdb+i-1] when order=Nag_ColMajor;
  • b[(i-1)×pdb+j-1] when order=Nag_RowMajor.
On entry: the n×r right-hand side matrix B.
On exit: if fail.code= NE_NOERROR, the n×r solution matrix X.
9: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax(1,n);
  • if order=Nag_RowMajor, pdbmax(1,nrhs).
10: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdb=value and nrhs=value.
Constraint: pdbmax(1,nrhs).
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
Element value of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be computed.

7 Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
(A+E) x^=b ,  
where
E1 = O(ε) A1  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κ(A) E1 A1 ,  
where κ(A) = A-11 A1 , the condition number of A with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
f07mpc is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04chc solves Ax=b and returns a forward error bound and condition estimate. f04chc calls f07mnc to solve the equations.

8 Parallelism and Performance

f07mnc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is approximately 43 n3 + 8n2r , where r is the number of right-hand sides.
The real analogue of this function is f07mac. The complex symmetric analogue of this function is f07nnc.

10 Example

This example solves the equations
Ax=b ,  
where A is the Hermitian matrix
A = ( -1.84i+0.00 0.11-0.11i -1.78-1.18i 3.91-1.50i 0.11+0.11i -4.63i+0.00 -1.84+0.03i 2.21+0.21i -1.78+1.18i -1.84-0.03i -8.87i+0.00 1.58-0.90i 3.91+1.50i 2.21-0.21i 1.58+0.90i -1.36i+0.00 )  
and
b = ( 2.98-10.18i -9.58+03.88i -0.77-16.05i 7.79+05.48i ) .  
Details of the factorization of A are also output.

10.1 Program Text

Program Text (f07mnce.c)

10.2 Program Data

Program Data (f07mnce.d)

10.3 Program Results

Program Results (f07mnce.r)